Both!!
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There are no geniuses. There are only people who have a lot of fun, interest and patience with a certain topic and follow their liking deeply. Main objective of school, university and life is to find your thing. And bacon.
(secx - tanx)^2 = (1-sinx)/(1+sinx) | * (1+sinx)
(secx - tanx)^2 = (1-sinx)/(1+sinx) | replc. secx, tanx
(1/cosx - sinx/cosx)^2 = (1-sinx)/(1+sinx) | combine lower
((1 - sinx)/cosx)^2 = (1-sinx)/(1+sinx) | ^2 each
(1-sinx)^2 / cosx^2 = (1-sinx)/(1+sinx) | replc. cosx
(1-sinx)^2 / (1-sinx^2) = (1-sinx)/(1+sinx) | binomi x2-y2=(x+y)/(x-y)
(1-sinx)^2 / ((1+sinx)(1-sinx)) = (1-sinx)/(1+sinx) | div by (1-sinx)
(1-sinx) / (1+sinx) = (1-sinx)/(1+sinx)
Took me quite a while, 35+ years later…
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I agree with you that math and objective thought are vitally important
The part I disagree with is dressing it up with words (secant?) and extraneous concepts that simply hide the fun part of the puzzle.
Trigonometry is a miracle. But it’s not a miracle that belongs in tenth grade. Problem solving is essential, and yet it’s omitted (by most curricula) except when necessary for testing purposes.
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Waitasec, in what friggin world is trigonometry inappropriate in a high school pre-calc class? What is your genius alternative, have everyone interested in STEM burn their first year playing catchup in college while going into a mortgage-sized debt? I was unfortunate enough to go to a high school that didn’t even offer calculus, but fortunately when I was playing catchup it only cost me four bucks a credit hour. But it still cost me a year.
If you’re not out of your goddam mind you’re doing a great impression of someone who is.
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Wolfram Alpha is your friend.
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42 because of math, maybe
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If we wanted to teach calculus to kids, starting today, with the tools we have now, without regard to how we did it post-Newton to a few people at a school like Harvard (did you know that they asked Galileo to be a guest professor?) how would we do it? I could teach calculus to ten year olds if they were into it. And I could do it without torturing the kids who were repulsed by the industrial methods of today’s teaching.
In the same vein, if we wanted to teach biology to kids, why wouldn’t we teach them chemistry first? The reason we teach biology before chemistry is that we figured out early biology hundreds of years before we figured out early chemistry, so there’s that. And yet, there’s not a lot of biology in learning chemistry, but there’s a ton of chemistry in learning biology…
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There is value in learning how to algebraically manipulate equations.
But I do agree with you in that we rarely teach the soul, the purpose of calculus. We don’t teach why Newton and Liebniz independently created calculus, or why the work is so important to us as a society today.
At one point a few years ago, I wanted to explore becoming a math teacher. (I didn’t; consulting was more lucrative. I’m not sure it was the right decision, but that’s for another day.) There’s a test you can take in NJ (the Praxis) to show competency in mathematics. The test, though, has little to do with being competent in the concepts of math, or in how to teach math. It’s a series of multiple choice questions in freshman/sophomore year math, presented in such a way that you need to be VERY good with a graphic calculator to get through the questions in time. I spent two months beforehand rehashing math, and learning the TI-80whatever it was. I was in the top 1%. I saw prospects open a brand-new calculator right before the test and knew they were doomed.
It was a test on learning how to use a calculator. I’ve long since forgotten how to use that calculator; I doubt I’d pass the test today, even though I use math all the time.
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There should be a trig(ger) warning on this post for those of us with math phobias.
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This particular problem may not be vital for everyday life, but it is the sort of thing that mathematicians need to be good at. It is a bit like practicing scales if you want to be a pianist. I doubt if I can convince those who don’t like maths, but here goes anyhow…
How do you prove there are an infinite number of primes? There is a proof by Euclid, though it is probably older than him. It goes like this…
Suppose you have a set of all the prime numbers you know about. Multiply them all together and add one. What divides exactly into that? All the prime numbers you knew about divide it, but leave remainder 1. So, either this is a new prime number, or there are two or more other numbers that divide into it exactly and aren’t in your set. Either way your set of prime numbers is never complete.
Ain’t that sweet?
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IMO trig becomes cool once you work with Taylor (and other) expansions of the trig functions into infinite polynomials.
trig is worthwhile for showing why 1 + e^(i*pi) = 0 if nothing else.
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I realize you were just using biology and chemistry as examples, but actually it makes more sense these days to teach chemistry first. Yes, traditional HS biology used to be mostly about anatomy and where equivalent organs were in worms, frogs, and people, and so needed little background, but these days they are incorporating a lot of molecular biology in there too. And to teach that you either have to have the students take chemistry first or have to waste half the class teaching the chemistry background they need.
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These books helped me through trig and calculus a lot.
I highly recommend them. They show the steps instead of giving you the back of the book answer.
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exactly @jhbadger we agree
and yet they teach biology first, at least in NY
I see what you did there.
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I went through an experimental high school curriculum that did physics > chemistry > biology. It was not great, though I’ll grant that may have been related to it being the first instance and it could have improved in subsequent iterations.
It started with a bunch of freshmen having to cram trigonometry basics in the first week of physics class, and then go on to complete that year of physics with no calculus at all. We were selected for the program by virtue of our strong math skills, and most of us rose to the occasion for at least the necessary trig—but I clearly remember being baffled by realizing I needed to find the exact slope of a curve at a specific point, and the teacher… expecting me to single-handedly re-invent calculus in my head, on demand, apparently? Like I identified that was what I needed to do, and asked the teacher, “how do you even do that?” His response was, “how do you think you do it?” It was like everyone involved had forgotten that “good at math” != “knows math,” and sometimes that’s actually an important distinction.
The physics class was still somehow the best part of the overall experience—it managed to go downhill from there to complete disengagement with the sciences on my part.
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From the perspective of people who used and enjoyed math at school, and/or still do, you’re coming in bafflingly hot here. Martin Gardner-type problems might be better for getting people interested in math, but that’s only valuable insofar as it germinates into actually doing math, and this is a nice, high-school-level exercise in the specific kinds of mental skills you need to maintain for that.
There’s nothing formulaic or arcane about the problem (sec is perhaps slightly old-fashioned, but whatever, if you have the trig identities to hand it’s just a symbol). You can’t solve this by applying some rote algorithm; you have to use imagination and be familiar with the way algebra moves – e.g., when you see an expression like (1-a2), you should guess that factoring it might get you somewhere.
If you want to critique education from a Seymour Papert angle, I’m here for that, but this example doesn’t make that case at all. Not all students need to learn math to this level, or do so at a prescribed age; but the idea that students who are learning math shouldn’t practice basic algebra is banañas.
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This would have taken thirty seconds if I hadn’t completely botched a term at step 2. But isn’t that always the case?
I will admit that the purpose of these things beyond an intellectual exercise failed to make much sense to me until my integral calculus course, which was presented as such an ingenious synthesis of everything that had come before – not that I have much need for it nowadays.
So much of what I learned, even the obscure bits, came in handy sooner or later. Except the rules of European handball. That’s still a mystery.
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